Reviewing the Classical and the de Bruijn Notation for -calculus and Pure Type Systems

Abstract

This article is a brief review of the type‐free λ‐calculus and its basic rewriting notions, and of the pure type system framework which generalises many type systems. Both the type‐free λ‐calculus and the pure type systems are presented using variable names and de Bruijn indices. Using the presentation of the λ‐calculus with de Bruijn indices, we illustrate how a calculus of explicit substitutions can be obtained. In addition, de Bruijn's notation for the λ‐calculus is introduced and some of its advantages are outlined.